We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Longerhans. Long term goals are to understand how the membrane dynamics interact with intracellular; events to regulate secretion and to generalize to other secretory cells and neurons. Our primary tool is the numerical solution of ordinary differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get analytical results in special cases. We work with both detailed biophysical models and simplified models which are more amenable to analysis. By selecting and combining features we can distill the essence of the phenomena, derive general principles, and facilitate communication with workers in related fields.